Journal of Dynamics and Differential Equations

, Volume 24, Issue 4, pp 897–925 | Cite as

Hopf Bifurcation in a Diffusive Logistic Equation with Mixed Delayed and Instantaneous Density Dependence



A diffusive logistic equation with mixed delayed and instantaneous density dependence and Dirichlet boundary condition is considered. The stability of the unique positive steady state solution and the occurrence of Hopf bifurcation from this positive steady state solution are obtained by a detailed analysis of the characteristic equation. The direction of the Hopf bifurcation and the stability of the bifurcating periodic orbits are derived by the center manifold theory and normal form method. In particular, the global continuation of the Hopf bifurcation branches are investigated with a careful estimate of the bounds and periods of the periodic orbits, and the existence of multiple periodic orbits are shown.


Reaction-diffusion equation Logistic equation Delayed and instantaneous density dependence Stability Local Hopf bifurcation Global Hopf bifurcation 

Mathematics Subject Classification (2010)

35K57 35R10 35B32 35B10 92D25 92D40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Azevedo K.A.G., Ladeira L.A.C.: Hopf bifurcation for a class of partial differential equation with delay. Funkcialaj Ekvacioj 47, 395–422 (2004)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Busenberg S., Huang W.: Stability and Hopf Bifurcation for a Population Delay Model with Diffusion Effects. J. Differ. Equ. 124, 80–107 (1996)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Chen, S., Shi, J.: Stability and Hopf bifurcation in a diffusive logistic population model with nonlocal delay effect. Submitted (2011)Google Scholar
  4. 4.
    Chow S.N., Hale J.K.: Methods of Bifurcation Theory. Springer, New York (1982)MATHCrossRefGoogle Scholar
  5. 5.
    Cooke, K.L., Huang, W.: A theorem of George Seifert and an equation with state-dependent delay. In: Fink, A.M., Miller, R.K., Kliemann, W. (eds.) Delay and Differential Equations, pp. 65–77. World Scientific, Singapore (1992)Google Scholar
  6. 6.
    Cushing J.M.: Integrodifferential Equations and Delay Models in Population Dynamics. Lecture Notes in Biomathematics, vol. 20. Springer, Berlin (1977)CrossRefGoogle Scholar
  7. 7.
    Dos Santos J.S., Bená M.A.: The delay efect on reaction-diffusion equations. Appl. Anal. 83, 807–824 (2004)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Faria T.: Normal form for semilinear functional differential equations in Banach spaces and applications. Part II. Disc. Cont. Dyn. Syst. 7(1), 155–176 (2001)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Faria, T., Huang, W.: Stability of periodic solutions arising from Hopf bifurcation for a reaction-diffusion equation with time delay. In: Differential Equations and Dynamical Systems (Lisbon, 2000), vol. 31, pp. 125–141. Fields Institute Communications, American Mathematical Society, Providence, RI (2002)Google Scholar
  10. 10.
    Faria T., Huang W., Wu J.: Smoothness of center manifolds for maps and formal adjoints for semilinear FDEs in general Banach spaces. SIAM J. Math. Anal. 34((1), 173–203 (2002)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Friedman A.: Remarks on the maximum principle for parabolic equations and its applications. Pac. J. Math. 8, 201–211 (1958)MATHGoogle Scholar
  12. 12.
    Friesecke G.: Convergence to equilibrium for delay-diffusion equations with small delay. J. Dyn. Differ. Equ. 5, 89–103 (1993)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gopalsamy K.: Stability and oscillations in delay differential equations of population. In: Mathematics and its Applications, vol. 74. Kluwer, Dordrecht (1992)Google Scholar
  14. 14.
    Green, D., Stech, H.: Diffusion and hereditary effects in a class of population models. In: Busenberg, S., Cooke, C. (eds.) Differential Equation and Applications in Ecology, Epidemics and Population Problems, pp. 19–28, Academic Press, New York (1981)Google Scholar
  15. 15.
    Gurney M.S., Blythe S.P., Nisbet R.M.: Nicholson’s bowflies revisited. Nature 287, 17–21 (1980)CrossRefGoogle Scholar
  16. 16.
    Hutchinson G.E.: Circular Causal Systems in Ecology. Ann. N. Y. Acad. Sci. 50, 221–246 (1948)CrossRefGoogle Scholar
  17. 17.
    Henry D.: Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840. Springer, Berlin (1981)Google Scholar
  18. 18.
    Huang W.: Global dynamics for a reaction-diffusion equation with time delay. J. Differ. Equ. 143, 293–326 (1998)MATHCrossRefGoogle Scholar
  19. 19.
    Krawcewicz W., Wu J.: Theory of degrees with applications to bifurcations and differential equations. In: Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1997)Google Scholar
  20. 20.
    Kuang Y., Smith H.L.: Global stability in diffusive delay Lotka-Volterra systems. Differ. Integr. Equ. 4, 117–128 (1991)MathSciNetMATHGoogle Scholar
  21. 21.
    Kuang Y., Smith H.L.: Convergence in Lotka-Volterra type diffusive delay systems without dominating instantaneous negative feedbacks. J. Aust. Math. Soc. B 34, 471–493 (1993)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Lenhart S.M., Travis C.C.: Global stability of a biological model with time delay. Proc. Am. Math. Soc. 96, 75–78 (1986)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Li W.T., Yan X.P., Zhang C.H.: Stability and Hopf bifurcation for a delayed cooperation diffusion system with Dirichlet boundary conditions. Chaos Solitons Fract. 38, 227–237 (2008)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    May R.M.: Time-delay versus stability in population models with two and three trophic levels. Ecology 54, 315–325 (1973)CrossRefGoogle Scholar
  25. 25.
    May R.M.: Stability and Complexity in Model Ecosystems. Princeton University Press, Princeton (1973)Google Scholar
  26. 26.
    Maynard-Smith J.: Models in Ecology. Cambridge University Press, Cambridge (1978)Google Scholar
  27. 27.
    Memory M.C.: Bifurcation and asymptotic behaviour of solutions of a delay-differential equation with diffusion. SIAM J. Math. Anal. 20, 533–546 (1989)MathSciNetMATHCrossRefGoogle Scholar
  28. 28.
    Miller R.: On Volterra’s population equation. SIAM J. Appl. Math. 14, 446–452 (1996)CrossRefGoogle Scholar
  29. 29.
    Morita Y.: Destabilization of periodic solutions arising in delay-diffusion systems in several space dimensions. Jpn. J. Appl. Math. 1, 39–65 (1984)MATHCrossRefGoogle Scholar
  30. 30.
    Nirenberg L.: A strong maximum principle for parabolic equations. Commun. Pure Appl. Math. 6, 167–177 (1953)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Pao C.V.: Dynamics of nonlinear parabolic systems with time delays. J. Math. Anal. Appl. 198, 751–779 (1996)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Parrot M.E.: Linearized stability and irreducibility for a functional differential equation. SIAM J. Math. Anal. 23, 649–661 (1993)CrossRefGoogle Scholar
  33. 33.
    Pazy A.: Semigroups of Linear Operators and Application to Partial Differential Equations. Springer, Berlin (1983)CrossRefGoogle Scholar
  34. 34.
    Ricklefs R.E., Miller G.: Ecology. W.H. Freeman, (1999)Google Scholar
  35. 35.
    Ruan, S.: Delay differential equations in single species dynamics. In: Delay Differential Equations and Applications (Marrakech, 2002), pp. 477–517. NATO Science Series II: Mathematics, Physics and Chemistry, vol. 205. Springer, New York (2006)Google Scholar
  36. 36.
    Seifert G.: On a delay differential equation for single specie population variations. Nonlinear Anal. 11, 1051–1059 (1987)MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Shi J., Shivaji R.: Persistence in reaction diffusion models with weak allee effect. J. Math. Biol. 52(6), 807–829 (2006)MathSciNetMATHCrossRefGoogle Scholar
  38. 38.
    Smith H.: An introduction to delay differential equations with applications to the life sciences. In: Texts in Applied Mathematics, vol. 57. Springer, New York (2011)CrossRefGoogle Scholar
  39. 39.
    Smoller J.: Shock Waves and Reaction-Diffusion Equations Grundlehren der Mathematischen Wissenschaften, vol. 258. Springer, New York (1983)Google Scholar
  40. 40.
    So J.W.-H., Yang Y.: Dirichlet problem for the diffusive Nicholson’s blowflies equation. J. Differ. Equ. 150, 317–348 (1998)MathSciNetMATHCrossRefGoogle Scholar
  41. 41.
    Su Y., Wei J., Shi J.: Bifurcation analysis in a delayed diffusive Nicholson’s blowflies equation. Nonlinear Anal. Real World Appl. 11, 1692–1703 (2010)MathSciNetMATHCrossRefGoogle Scholar
  42. 42.
    Su Y., Wei J., Shi J.: Hopf bifurcation in a reaction-diffusion population model with delay effect. J. Differ. Equ. 247, 1156–1184 (2009)MathSciNetMATHCrossRefGoogle Scholar
  43. 43.
    Travis C., Webb G.: Existence and stability for partial functional differential equations. Trans. Am. Math. Soc. 200, 395–418 (1974)MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Wu J.: Theory and Applications of Partial Functional-Differential Equations. Springer, New York (1996)MATHCrossRefGoogle Scholar
  45. 45.
    Wu J.: Symmetric functional differential equations and neural networks with memory. Trans. Am. Math. Soc. 350, 4799–4838 (1998)MATHCrossRefGoogle Scholar
  46. 46.
    Yan X., Li W.: Stability of bifurcating periodic solutions in a delayed reaction-diffusion population model. Nonlinearity 23, 1413–1431 (2010)MathSciNetMATHCrossRefGoogle Scholar
  47. 47.
    Yoshida K.: The Hopf bifurcation and its stability for semilinear diffusion equations with time delay arising in ecology. Hiroshima Math. J. 12, 321–348 (1982)MathSciNetMATHGoogle Scholar
  48. 48.
    Zhou L., Tang Y., Hussein S.: Stability and Hopf bifurcation for a delay competition diffusion system. Chaos Solitons Fract. 14, 1201–1225 (2002)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinPeople’s Republic of China
  2. 2.Department of Applied MathematicsUniversity of Western OntarioLondonCanada
  3. 3.Department of MathematicsCollege of William and MaryWilliamsburgUSA

Personalised recommendations